http://www.ck12.org Chapter 8. Right Triangle Trigonometry
8.9 Inverse Trigonometric Ratios
Here you’ll learn how to use inverse trigonometric ratios to solve for missing angles in right triangles.
What if you were told that the longest escalator in North America is at the Wheaton Metro Station in Maryland
and is 230 feet long and is 115 ft high? What is the angle of elevation,x◦, of this escalator? After completing this
Concept, you’ll be able use inverse trigonometry to answer this question.
Watch This
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Click image to the left for more content.CK-12 Foundation: Chapter8InverseTrigonometricRatiosA
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Click image to the left for more content.James Sousa:Introduction toInverseTrigonometric Functions
Guidance
The wordinverseis probably familiar to you. In mathematics, once you learn how to do an operation, you also learn
how to “undo” it. For example, you may remember that addition and subtraction are considered inverse operations.
Multiplication and division are also inverse operations. In algebra you used inverse operations to solve equations
and inequalities. When we apply the word inverse to the trigonometric ratios, we can find the acute angle measures
within a right triangle. Normally, if you are given an angle and a side of a right triangle, you can find the other two
sides, using sine, cosine or tangent. With the inverse trig ratios, you can find the angle measure, given two sides.
Inverse Tangent:If you know the opposite side and adjacent side of an angle in a right triangle, you can use inverse
tangent to find the measure of the angle. Inverse tangent is also called arctangent and is labeled tan−^1 orarctan. The
“-1” indicates inverse.
Inverse Sine:If you know the opposite side of an angle and the hypotenuse in a right triangle, you can use inverse
sine to find the measure of the angle. Inverse sine is also called arcsine and is labeled sin−^1 orarcsin.
Inverse Cosine:If you know the adjacent side of an angle and the hypotenuse in a right triangle, you can use inverse
cosine to find the measure of the angle. Inverse cosine is also called arccosine and is labeled cos−^1 orarccos.
Using the triangle below, the inverse trigonometric ratios look like this: